Albert Satorra

Bio: Albert Satorra is an academic researcher from Pompeu Fabra University. The author has contributed to research in topics: Estimator & Goodness of fit. The author has an hindex of 37, co-authored 121 publications receiving 14645 citations. Previous affiliations of Albert Satorra include Barcelona Graduate School of Economics & BI Norwegian Business School.

A Scaled Difference Chi-square Test Statistic for Moment Structure Analysis

Abstract: A family of scaling corrections aimed to improve the chi-square approximation of goodness-of-fit test statistics in small samples, large models, and nonnormal data was proposed in Satorra and Bentler (1994). For structural equations models, Satorra-Bentler's (SB) scaling corrections are available in standard computer software. Often, however, the interest is not on the overall fit of a model, but on a test of the restrictions that a null model sayM 0 implies on a less restricted oneM 1. IfT 0 andT 1 denote the goodness-of-fit test statistics associated toM 0 andM 1, respectively, then typically the differenceT d =T 0−T 1 is used as a chi-square test statistic with degrees of freedom equal to the difference on the number of independent parameters estimated under the modelsM 0 andM 1. As in the case of the goodness-of-fit test, it is of interest to scale the statisticT d in order to improve its chi-square approximation in realistic, that is, nonasymptotic and nonormal, applications. In a recent paper, Satorra (2000) shows that the difference between two SB scaled test statistics for overall model fit does not yield the correct SB scaled difference test statistic. Satorra developed an expression that permits scaling the difference test statistic, but his formula has some practical limitations, since it requires heavy computations that are not available in standard computer software. The purpose of the present paper is to provide an easy way to compute the scaled difference chi-square statistic from the scaled goodness-of-fit test statistics of modelsM 0 andM 1. A Monte Carlo study is provided to illustrate the performance of the competing statistics.

Complex Sample Data in Structural Equation Modeling

Abstract: Large-scale surveys using complex sample designs are frequently carried out by government agencies. The statistical analysis technology available for such data is, however, limited in scope. This study investigates and further develops statistical methods that could be used in software for the analysis of data collected under complex sample designs. First, it identifies several recent methodological lines of inquiry which taken together provide a powerful and general statistical basis for a complex sample, structural equation modeling analysis. Second, it extends some of this research to new situations of interest. A Monte Carlo study that empirically evaluates these techniques on simulated data comparable to those in largescale complex surveys demonstrates that they work well in practice. Due to the generality of the approaches, the methods cover not only continuous normal variables but also continuous nonnormal variables and dichotomous variables. Two methods designed to take into account the complex sample structure were

Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic.

Abstract: A scaled difference test statistic \(\tilde{T}{}_{d}\) that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (Psychometrika 66:507–514, 2001). The statistic \(\tilde{T}_{d}\) is asymptotically equivalent to the scaled difference test statistic \(\bar{T}_{d}\) introduced in Satorra (Innovations in Multivariate Statistical Analysis: A Festschrift for Heinz Neudecker, pp. 233–247, 2000), which requires more involved computations beyond standard output of SEM software. The test statistic \(\tilde{T}_{d}\) has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic \(\bar{T}_{d}\) that avoids negative chi-square values.

Ensuring Positiveness of the Scaled Difference Chi-square Test Statistic

Abstract: A scaled difference test statistic T_tildad that can be computed from standard software of structural equation models (SEM) by hand calculations was proposed in Satorra and Bentler (2001). The statistic T_tildad is asymptotically equivalent to the scaled difference test statistic T_hatd introduced in Satorra (2000), which requires more involved computations beyond standard output of SEM software. The test statistic T_tildad has been widely used in practice, but in some applications it is negative due to negativity of its associated scaling correction. Using the implicit function theorem, this note develops an improved scaling correction leading to a new scaled difference statistic T_hatd that avoids negative chi-square values.

Cutoff criteria for fit indexes in covariance structure analysis : Conventional criteria versus new alternatives

Abstract: This article examines the adequacy of the “rules of thumb” conventional cutoff criteria and several new alternatives for various fit indexes used to evaluate model fit in practice. Using a 2‐index presentation strategy, which includes using the maximum likelihood (ML)‐based standardized root mean squared residual (SRMR) and supplementing it with either Tucker‐Lewis Index (TLI), Bollen's (1989) Fit Index (BL89), Relative Noncentrality Index (RNI), Comparative Fit Index (CFI), Gamma Hat, McDonald's Centrality Index (Mc), or root mean squared error of approximation (RMSEA), various combinations of cutoff values from selected ranges of cutoff criteria for the ML‐based SRMR and a given supplemental fit index were used to calculate rejection rates for various types of true‐population and misspecified models; that is, models with misspecified factor covariance(s) and models with misspecified factor loading(s). The results suggest that, for the ML method, a cutoff value close to .95 for TLI, BL89, CFI, RNI, and G...

Comparative fit indexes in structural models

Abstract: Normed and nonnormed fit indexes are frequently used as adjuncts to chi-square statistics for evaluating the fit of a structural model A drawback of existing indexes is that they estimate no known population parameters A new coefficient is proposed to summarize the relative reduction in the noncentrality parameters of two nested models Two estimators of the coefficient yield new normed (CFI) and nonnormed (FI) fit indexes CFI avoids the underestimation of fit often noted in small samples for Bentler and Bonett's (1980) normed fit index (NFI) FI is a linear function of Bentler and Bonett's non-normed fit index (NNFI) that avoids the extreme underestimation and overestimation often found in NNFI Asymptotically, CFI, FI, NFI, and a new index developed by Bollen are equivalent measures of comparative fit, whereas NNFI measures relative fit by comparing noncentrality per degree of freedom All of the indexes are generalized to permit use of Wald and Lagrange multiplier statistics An example illustrates the behavior of these indexes under conditions of correct specification and misspecification The new fit indexes perform very well at all sample sizes

On the evaluation of structural equation models

Abstract: Criteria for evaluating structural equation models with latent variables are defined, critiqued, and illustrated. An overall program for model evaluation is proposed based upon an interpretation of converging and diverging evidence. Model assessment is considered to be a complex process mixing statistical criteria with philosophical, historical, and theoretical elements. Inevitably the process entails some attempt at a reconcilation between so-called objective and subjective norms.

lavaan: An R Package for Structural Equation Modeling

Abstract: Structural equation modeling (SEM) is a vast field and widely used by many applied researchers in the social and behavioral sciences. Over the years, many software packages for structural equation modeling have been developed, both free and commercial. However, perhaps the best state-of-the-art software packages in this field are still closed-source and/or commercial. The R package lavaan has been developed to provide applied researchers, teachers, and statisticians, a free, fully open-source, but commercial-quality package for latent variable modeling. This paper explains the aims behind the development of the package, gives an overview of its most important features, and provides some examples to illustrate how lavaan works in practice.

Missing data: Our view of the state of the art.

Abstract: Statistical procedures for missing data have vastly improved, yet misconception and unsound practice still abound. The authors frame the missing-data problem, review methods, offer advice, and raise issues that remain unresolved. They clear up common misunderstandings regarding the missing at random (MAR) concept. They summarize the evidence against older procedures and, with few exceptions, discourage their use. They present, in both technical and practical language, 2 general approaches that come highly recommended: maximum likelihood (ML) and Bayesian multiple imputation (MI). Newer developments are discussed, including some for dealing with missing data that are not MAR. Although not yet in the mainstream, these procedures may eventually extend the ML and MI methods that currently represent the state of the art.